Question

Donations. Organizers of a televised fundraiser know from past experience that most people donate small amounts \((\$ 10-\$ 25)\), some donate larger amounts \((\$ 50-\$ 100)\), and a few people make very generous donations of \(\$ 250, \$ 500\), or more. Historically, pledges average about \(\$ 32\) with a standard deviation of \(\$ 54\). a) If 120 people call in pledges, what are the mean and standard deviation of the total amount raised? b) What assumption did you make in answering this question?

Short answer

The mean total is $3840, and the standard deviation is approximately $592.82. The assumption is that each pledge is an independent and identically distributed random variable.

Step-by-step solution

Understand the Question

We are given historical data about donations and asked to calculate the mean and standard deviation for the total amount raised from 120 pledges. We also need to identify any assumptions made.

Calculate Mean of Total Donations

The mean donation per person is given as \( \\(32 \). For 120 people, the mean of the total donations can be calculated by multiplying the mean per person by the number of people: \[\mu_{{\text{total}}} = 120 \times 32 = 3840\]Thus, the mean of the total amount raised is \( \\)3840 \).

Calculate Standard Deviation of Total Donations

The standard deviation of donations per person is \( \\(54 \). The standard deviation of the total donations for 120 people can be calculated using the formula for the sum of independent random variables:\[\sigma_{{\text{total}}} = \sqrt{120} \times 54\]Calculate this to find:\[\sigma_{{\text{total}}} \approx 592.82\]Thus, the standard deviation of the total amount raised is approximately \( \\)592.82 \).

Identify Assumption

To solve this problem, we assume that each pledge is an independent random event and the distribution of donations for each donor is identical, which allows us to use the mean and standard deviation to calculate the parameters for the total donation.

Key concepts

Mean Calculation

Understanding the mean calculation is essential in statistics, where it represents the average value of a dataset. In our exercise, the mean donation amount is given, but how do we find the mean for a group? This is quite simple. If each donor donates an average of \(\(32\), with 120 donors, we multiply the mean donation by the number of donors to find the mean of the total donations.
This means:

• Mean per person: \(32\)
• Number of donors: 120

To calculate the mean of the total:\[\mu_{\text{total}} = 120 \times 32 = 3840\]Hence, on average, they expect to raise \(\)3840\). This is an example of how averages can be used to predict total outcomes based on individual contributions.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means that most of the numbers are close to the mean, while a high standard deviation indicates that the numbers are spread out over a wider range.
In our fundraiser scenario, each donor's contribution has a standard deviation of \(\(54\). To find the standard deviation of the total donations from 120 donors, we apply the formula for the standard deviation of the sum of independent random variables:

• Standard deviation per person: \(54\)
• Number of donors: 120

The calculation is:\[\sigma_{\text{total}} = \sqrt{120} \times 54 \approx 592.82\]This indicates that while the average total donation is \(\)3840\), it may typically vary by around \($592.82\). This helps predict the possible range in total fundraising outcomes.

Random Variables

In statistics, a random variable represents a numerical outcome of a random phenomenon. They are essential for modeling situations where outcomes are uncertain. In our case, each donation can be considered a random variable with its probability distribution based on past data.
Random variables allow us to use statistical formulas to make predictions for the group as a whole, like calculating the mean total or standard deviation. Each donor's pledge acts as an independent random variable.

• Each donation: a random variable
• Assumes independent outcomes

Treating pledges as random variables lets us apply statistical methods that assume independence and identical distribution. This lets us combine individual statistics to estimate parameters for total donations.

Assumptions in Statistics

In many statistical problems, making certain assumptions allows us to use known formulas and methods to find solutions. The assumptions we make can significantly affect our conclusions and predictions.
For the fundraiser scenario in question, we assume that all donations are independent and identically distributed random variables. This means:

• Each donation is independent of others
• Donations have a similar distribution pattern

These assumptions let us apply straightforward calculations for the mean and standard deviation. It is important to note, however, that if these assumptions do not hold in real-world data, our predictions could be inaccurate. Therefore, understanding underlying assumptions is crucial for proper application of statistical methods.